Position and Displacement — Question 2

Question 2

A spatial route telescopes to its endpoints

A drone follows straight legs through A $(1,2,0)$ m, B $(4,6,0)$ m, C $(4,6,12)$ m, then D $(1,2,12)$ m in that order. The z axis is upward.

1. Find each leg displacement and the net displacement.

2. Find the total distance and compare it with the net magnitude.

3. If B and C are exchanged in the visiting order, which result remains fixed? Find the new distance.

Question 2 - Solution

A spatial route telescopes to its endpoints

(a) Subtract successive positions.

$$\Delta\vec r=(3,4,0)+(0,0,12)+(-3,-4,0)=\boxed{(0,0,12)\,\mathrm m}$$ $\Delta\vec r_{AB}=(3,4,0)$ m, $\Delta\vec r_{BC}=(0,0,12)$ m and $\Delta\vec r_{CD}=(-3,-4,0)$ m. The sum is $(0,0,12)$ m, equal to $\vec r_D-\vec r_A$; intermediate positions cancel.

(b) Sum leg lengths for traveled distance.

$$s=\sqrt{3^2+4^2}+12+\sqrt{(-3)^2+(-4)^2}=22\,\mathrm m>12\,\mathrm m$$ The lengths are 5, 12 and 5 m, so distance is 22 m. The net displacement magnitude is 12 m upward. Horizontal travel cancels in the endpoint difference but remains part of the traveled distance.

(c) Keep endpoints while changing the route.

$$s'=\sqrt{3^2+4^2+12^2}+12+\sqrt{3^2+4^2+12^2}=38\,\mathrm m$$ The order A–C–B–D has leg lengths 13, 12 and 13 m, totaling 38 m. Its displacement remains $(0,0,12)$ m. A route change cannot alter the endpoint displacement when the start and finish are fixed.